An exactly solvable model in predictive relativistic mechanics. I

A NNALES DE L ’I. H. P., SECTION A

V. I ^RANZO J. L ^LOSA F. M ^ARQUÉS A. M ^OLINA

An exactly solvable model in predictive relativistic mechanics. I

Annales de l’I. H. P., section A, tome 35, n

^o

1 (1981), p. 1-15

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1

An exactly solvable model

in Predictive Relativistic Mechanics. I

V. IRANZO, ^J. LLOSA, F.

MARQUÉS

and A. MOLINA

Departament ^{de Fisica} Teorica, Universitat de Barcelona (*)

Vol. XXXV, ^n° 1, 1981, Physique ^’ théorique. ^’

ABSTRACT. 2014 We present ^a

family

^of hamiltonian models for

N-particle

systems ⁱⁿ Predictive Relativistic Mechanics. The relation between

posi-

tion and

velocity

variables and the

non-physical

^{canonical ones} is derived.

The case of a harmonic oscillator

potential

is studied in detail. We compare

our model with some

singular lagrangian

systems

already appeared

ⁱⁿ

the literature.

1. INTRODUCTION

This is the first of a series of _papers about a class of systems ^of

particles interacting

^{at a} distance and their

quantization.

^The interest of relativistic action-at-a-distance theories has

recently

grown

[7] ] [2] ] [3] ] [4] ] [J] ] [6].

From the theoretical

viewpoint

this is due to the fact that the simultaneous character of the interaction has been made

compatible

with Poincare invariance

by

Predictive Relativistic Mechanics.

Also,

hamiltonian _sys-

tems with constraints have been better understood. These systems ^are

only

^{defined on a}

submanifold

^{of the}

phase

space and this enables one to circumvent the no-interaction theorem. The connexion between both

(*) ^{Postal address:} Diagonal, 647, Barcelona-28, Spain.

Annales de l’Institut Henri Poincaré-Section A-Vol. XXXV, 0020-2339/1981/1/$ 5,00/

(Ç) Gauthier-Villars

2 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA}

formalisms,

Predictive Relativistic Mechanics and

Singular Lagrangian Theory

^{has been}

recently

established

by

^{the authors}

[7] ] [8 ].

From a

phenomenological viewpoint

^{the use of covariant harmonic}

oscillator models has allowed to obtain the mass spectrum ^{of hadrons} and has

provided

^a theoretical _way to

explain

^the

quark-confine-

ment

[9] ] [10 ].

Before the work of Droz-Vincent

[5] ] only perturbative

solutions of

physically interesting problems

^{were known}

[7] ] [79] ] [20] ]

within the framework of Predictive Relativistic Mechanics.

We present ^{here a}

family

^of

exactly

solvable models which includes

as a

particular

^{case for two}

particles

the Droz-Vincent system.

Although

^{we do not} believe that this kind of

simple

^{models can describe}

the interaction between

quarks,

this will be

only possible

^{in the framework}

of a

relativistic

second

quantization. Nevertheless,

^the

clearness,

^{the sim-}

plicity

^{and the exact}

solvability

of these models

permit

^a better under-

standing

of the main features of

quark

interaction.

In sec. 2 we present ^a short reviews of Predictive Relativistic Mecha- nics and the

general

characteristics of ^our model. It is

inspired

^{on Dirac’s}

idea

[77] ]

^of

distinguishing

the kinematic generators ^{from the}

dynamic

ones among those of the Poincare

Group.

^In our case we make the above mentioned distinction _among the generators ^{of the}

Complete Symmetry Group

^{which is an} abelian extension of the Poincare

Group.

In sec. 3 we present the model in detail. In order to circumvent the No- Interaction Theorems

[72] ] [7~] ]

^we deal with a

non-physical

^canonical

coordinates. The relation between the

position

^and

velocity

^variables

and the canonical ones is derived in sec. 4. In sec. 5 we present ^some

appli-

cations of the model: harmonic oscillator

potential,

^free

particles

system and

comparison

^{with some}

singular lagrangian

^models.

2. PREDICTIVE HAMILTONIAN SYSTEMS

A Predictive Poincare invariant system ^{is a} second order differential system

which describes the

dynamics

^{of N}

particles.

The acceleration

0~

^must

satisfy [7~] ] [7~] :

Annales de l’Institut henri Poincaré-Section A

where the summation convention holds for c, ^{... ...}

indices,

^{and the metric is taken}

(2014+++).

Equation (2.3)

^{exhibits that} 6a is the proper time of the

particle

^{« a »}

1 appart ^{from a}

multiplicative

^{constant which we} shall take

equal

^{to 2014.}

If _~a were the proper

time,

^{it would}

yield

^~

and the coordinates

(x, 7~)

^{would no}

^longer

^be

independent.

^{This would}

obstruct the construction of a halmitonian formalism. On the contrary, the choice of mentioned above allows

~a

^{to take on} any value

which, according

^to eq.

(2.2),

^{will be an}

integral

^{of motion.}

Equations (2.3)

show that the solution _xa of system

(2.1) only depends

on the parameter 6a.

Furthermore, they

guarantee ^the

integrability

^of

the system.

Equations (2.3)

^{can also} be written as :

The invariance of

(2 .1 )

^{under the Poincare} group

implies

^{that the func-}

tions

8a(x, ~c)

^{behave as} four-vectors invariant under translations. This is e uivalent ^to

where :

are the infinitesimal generators of the Poincare

Group.

Equations (2.4)

^and

(2.5)

exhibit that

Ha, PJl’

span ^a realization of an abelian extension of the Poincare

algebra [16 ].

The associated trans- formation

group ~

^{is the direct}

product

^{of the Poincare} group

(purely kinematic)

^{and the}

dynamical

group [RN.

A hamiltonian formalism for the system

(2 .1 )

is determined

by

^a sym-

plectic

^{form Q on} invariant under J 8&#x3E;

AN.

As it is well

known,

^{a function}

f

^can be associated to each field A that leaves Q invariant. This function is defined

by [16 ].

This function is defined at least

locally

^{on and it is}

unique

except for an

arbitrary

^{additive constant.}

4 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA}

In

particular,

the functions associated to and

Ha

^{are the linear} momentum, ^the

angular

^{momentum and the}

hamiltonians, respectively.

This is the _way to follow in order to

quantize

^{a classical} system like

(2.1) [17].

^{We must}

point

^{out that this}

procedure

^involves pertur- bative solutions as in the cases of

electromagnetic

interaction and of short range scalar and vector interactions

[18 ].

We will

follow, however,

Droz-Vincents

approach [5 ].

^{Instead of}

constructing

^a hamiltonian formalism for a

given

interaction

(e.

g. elec-

tromagnetism...),

^we

display

^some

exactly

solvable models which serve us to know better the structure of these theories. We also want some of these models to describe the interaction between

quarks.

We work in an

adapted

canonical coordinates system

p~)

^ofQ

^{[16 ],}

i. e. : these coordinates

satisfy

^the

following

conditions :

i ) q~

^and

pb (a,

^{b =}

1,

^...,

N)

^are four-vectors

and qa

behaves like

~

under translations.

ii)

^the

sympletic

^{form Q can be}

expressed

^{as :}

or,

equivalently,

^the Poisson bracket associated ^to Q is such that :

Then, if

Pu

^and

J 111-

^{are the} functions associated

by

eq.

(2.7)

^to the gene- rators

Pl1

^and

of !J&#x3E;,

^we

immediately

obtain from the condition

(i)

^{that :}

Therefore,

^the

dynamical

aspects ^{of the} system ^{will be} present ^{in the} functions

Ha(q, p) (associated

^to

Ha)

^{and in the functions}

x~(q, p), ~(~, p)

which relate the

position

^and

velocity

^{variables to the canonical coor-}

dinates and momenta.

Hence,

^a model will consist in

giving

^{N function}

p) satisfying

^the

following

conditions :

i)

^each

Ha( q, p)

is invariant under the Poincare _group

Both conditions guarantee that eq.

(2.4)

^and

(2.5)

^{will be fulfilled}

by

the

generators Ha

associated to

Ha(q, p) by (2.7).

After that several

problems

still remain. First we have to derive the

relationship

between the

adapted

canonical coordinates

(q, p)2014which

have ^not any

physical significance and

^the

position

^and

velocity

variables

~).

^{As far as we do not} know these relations our hamil- tonian model will neither ^be able to

predict anything

^{nor to be}

compared

with other

physical

^models.

l’Institut Henri Poincaré-Section A

Takin

^{into account} eq .

( 2 .1 ),

the functions x 03B1 ^must

satisfy :

This means that the evolution of

xa depends only

^{on the} parameter 6a, unlike the canonical

coordinates q a

^which

depend

^{on the whole set of} parameters ... ,

6N~.

.

If we can solve _eq.

(2.11)

^{we will know the relation}

p).

^{As we shall}

see later a

good

^set of initial data will

permit

^{us to collect one} among the

infinity

of solutions of

(2.11).

Secondly, according

^to

(2.2) {H~ ought

^{to be an inte-}

gral

of motion. It is

extremely complicated

^to

impose

this condition.

However we can circumvent this

difficulty by assuming

^{that the field}

associated

trough

^{Q to}

p)

^{is not}

equal

^to

Ha

^but

only proportional.

That is to say, if we write :

the parameter 03C4a ^{is no}

longer equal

^to the proper time

Now, given

^{a solution}

xa

=

p) (a

⁼

1,

^...,

N)

^of eq.

(2.10),

^{we need}

only

^to

require

^the

change

of variables

to be invertible.

There is at last one

problem

left : what is the relation between

(x, x)

and

(x, 7r)? As x a and 03C0 a

^are

parallel

^we

only

^{need the} ratio between their

lengths

^{in order to define}

precisely

^the

change

of variables. This can be done

by giving

^a fixed value to each

integral

^{of motion ...,}

HN . Similarly

^{to the case of free}

particles

^{or of}

separable

interaction

[7~] ]

we shall take

The relation between the parameters 6Q

and 03C4a

^{can be}

easily

^{obtained :}

where C. I. means the set of initial conditions which determine the

trajec-

tories. The

change

^of parameter ^{can be now} obtained from

(2.15) by

^a

quadrature.

3. A FAMILY OF MODELS

We shall choose the hamiltonians

Ha

^{with the}

following

^{form :}

where :

Ta

^{is a}

quadratic

^function

of pb,

^{V is a} Poincare invariant function and it is the same for all a.

6 V. IRANZO. ^J. LLOSA, ^F. MARQUES AND A. MOLINA

For the sake of

simplicity

^we shall also

require

^{that both}

Ta

^{and V are} well behaved under

particle interchange.

^{That is to} say, if ^{6 E}

SN (the symmetric

group of N

elements),

^{then :}

This condition

implies

^{that :}

where

03B11, 03B21,

7i

and 03B41

1 are

arbitrary

^constants.

Instead of

using

^{the variables}

p~)

^{it is more suitable to}

^perform

^the

canonical transformation :

c ~ r, , ... , ^{i .}

The new variables P, y03BDAm z03BBB ^are translation invariant.

Hence, V(q, p)

^must

be some function of the

following

^scalar

varia-bles:

where

ZB

^{are the}

projections of y A, Za perpendicular

^{to Pu :}

In terms of the new

variables,

eq.

(3.2) yields :

where,

^{in orden to}

simplify

^the

notation,

^{we have} introduced :

and (x,

~3, y, ~

^are

arbitrary

^constants.

The functions

Ha

^{that we have chosen must}

satisfy

^the

predictivity equations (2.10).

^{As the} hamiltonians

Ha

^are well behaved under

particle interchange

^{it suffices to}

require : {

^{= 0 A =}

^{2, 3,}

^{..., N.}

Using (3 . 6)

these conditions

yield :

Annales de ^I Henri Poincaré-Section A

Taking into account that V depends ^{on the variables}

(3 . 4)

^{we obtain :}

The

simplest

^{models are} those with b ⁼ 0. Otherwise the differential system

(3. 7)

^becomes

extremely complicated.

^{For these}

simplified

^models

eq.

(3 . 7) yields :

Hence,

^{if we choose 5 = 0 and}

V(P2, PyA, AB, AB, AB)

^the pre-

dictivity

^condition

(2.9)

^is

automatically

^fulfilled.

Since V does not

depen

^{on the functions}

PyA

^are

integrals

^of

motion.

And,

^{as p2 is also an}

integral

^{of motion the}

products (PpJ

^are

conserved

quantities,

^too.

Other

integrals

^{of motion are} those associated to the generators ^{of the} Lorentz group :

which can be

expressed

^{as :}

with :

where : M ⁼

( - P2) 1 ~2

^{and is the} four-dimensional Levi-Civita tensor.

Analogously

^to the canonical transformation

(3 . 3)

^we

perform

^a

change

of parameters :

It

immediately

follows from _eq.

(3.12&#x26;)

^that

YÀ

^and

zA

^{do not}

depend

on

ÀB

^and

they

^are

solely

functions of ~,.

Therefore,

^the transverse internal motion of the system

(i.

^e.: the evolution of the relative coordinates and

Vol. 1-1981.

8 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA}

momenta in the

orthogonal hiperplane

^{to is}

governed by

^the

ordinary

second order differential system :

The

potential

^{V can}

depend

^on

~ 2~

^{but also on p2 and}

However,

as p2 and

PyA

^are

integrals

^{of motion}

they

^{can be}

regarded

^as parameters in order to

integrate

^the system

(3.13). Therefore,

eqs.

(3.13)

^{allows us to}

obtain

yA(~,;

^C.

I.)

^and

2À(À;

^C.

I.) independently

^{of the} evolution of the other variables.

Finally,

the evolution

equations

^of

(PX)

^and

(PzA) yield :

~ A

If V does not

depends

^{on p2 and}

PyA,

^the system

(3 .13)

^{would be imme-}

diately integrable,

^{because p2 and}

PyA

^{are conserved}

quantities.

^{As we}

shall see

later,

^{if we know}

(PX)

^and

(PzA)

^{in terms}

of ..1, ~,A

^{and the initial}

conditions then we can

easily integrate

^the

position equations (2 .11 ).

Therefore we shall consider hereafter

only potentials

of the form :

zc).

In this case,

by integration

^of eq.

(3 .14)

^{we obtain :}

The hamiltonians to be considered hereafter are therefore:

4. INTEGRATION OF THE POSITION

EQUATIONS

In order to

complete

^{the model}

proposed

in the last section we must

integrate

^the

position equation (2.11).

By analogy

^{with the}

two-particle

^model of Droz-Vincent

[5] and

^with

Annales de l’Institut Henri Poincaré-Section A

some

singular lagrangians [2] ^] [3] ] [4] we

shall take the

following

^initial

conditions :

The Frobenius theorem

guarantees

^{us that} there exists a

unique

solution of

(2 .11 ) satisfying

^{the initial} conditions

(4 .1 ).

^{Given one}

point

there exists a

unique integral cpa(~1, ..., ’~N),

...,

TN)

^{of the}

generalized

^Hamilton

equations depending

^{on N} para- meters :

Satisfying ~(0.. ~ 0) = ~

^...,

0)

⁼

p~.

By

variation of all the

parameters

^{but one} za, this

integral

submanifold

permits

^{us to} go from

Q

^to

Qa

⁼

p~)

where :

bza

^{= 0 and are} suitable variation of the

parameters.

From eq.

(2.11) p)

^{is not altered}

by

^a

change

^of

03C4a,(a’ ~ a).

^Hence

xa(Q)

^is

equal

^{to and}

using

eq.

(4.1)

^{we have:}

in terms of

Q

⁼

(q~(Ü)’

Therefore,

^{if we can} express in terms of

Q

⁼

(~S, p~)

^{the inte-}

gration

^of

(2.11)

^{with the} initial conditions

(4.1)

^{will be concluded.}

From eq.

(3.3), (3.11)

^and

(4.3)

^{we obtain}

where the subindex has been omitted from

Ku, M,

^{pv and because}

they

^are

integrals

^{of motion.}

Writing

eq.

(3.15)

^at

Q

^and

Qm substracting

^{them and}

taking

^{into account}

eq.

(4.1),

^{we have :}

where

~~,A, ~~,

^are

^expressed

^{in terms of the}

^{(N - 1)}

parameters

by (3.12).

Vol. XXXV, ^n° 1-1981.

10 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA}

We can find the unknowns

(PX)

^and

U

^{from eq.}

(4 . 5)

^{and we obtain :}

where the subindex

Q

^is understood in the

quantities

^{on the}

right

^{hand side.}

Now the

quantities (~ 2014

^{are still to be found. Let us write}

^(3.13)

^{as :}

I.

C.), ~;

^I.

C.)}

^{be the}

general integral

^of

(4 . 8). ^Then,

it is

quite

clear that :

where 5x is

given by

eq.

(4.7).

Substituting

eq.

(4.9)

^and

(4.6)

in eq.

(4.4)

^the

integration

^{of the}

posi-

tion

equations (2.11)

^is

complete.

^{We have to}

point

^{out that}

only

^the

term

(~ 2014 ^depends

^{on the}

potential.

^Since

^Q

^is

arbitrary,

^{the result}

is

general,

^so the indices

Q

^and

(0)

^{will be}

dropped

^{from now on.}

The

change

of variables

( q, p)

^--+

(x, x)

^{can be}

finally

^{written as :}

were the subindex

(0)

^is

dropped

^from

(4.9).

If this

change

of variables is to be invertible

and acb

^{and pl1 well}

^behaved,

the followins conditions must be also satisfied :

In any ^{case ’ we can} choose o

~,

y and ⁰ V such that the conditions above hold ⁰ in

(TM4)N

except,

perhaps,

^{in a set of measure ’ zero.}

Annales de l’Institut Henri Poincare-Section A

5. APPLICATIONS a~ Harmonic oscillator.

In order to

explain

^the interaction between

quarks,

^different

models of relativistic harmonic oscillators ^have

appeared

ⁱⁿ the litera- ture

[27] ] [22] ] [5] ] [6 ].

^{We shall now} present ^{a model}

of N-particle

^har-

monic

oscillator,

^{which for N = 2 includes the} model of Droz-Vincent

[5] ]

mentioned above as a

particular

^{case. We shall use the}

potential:

Then,

^the

equations

of motion for the relative transverse coordinates and momenta can be written as :

Then,

^{we obtain the}

following eq uation

In other

words,

^{all the relative} transverse coordinates oscillate with the same

frequency

^{úJ =}

NN03B3K.

On the other

hand,

eq.

(3.15)

^shows

that the component

parallel

^{to pl1 of these} coordinates

depends linearly

on za.

To

completely

solve this model we must derive the

change

^of

variables

( q, p)

^-~

(x, x).

^From

(4 . 9)

^and

(5 . 3)

^{we obtain}

easily :

where ~~. is

given by (4. 6).

The introduction of the results

(5 . 4)

ⁱⁿ eq.

(4 .10)

a

and

(4. .11 )

determines the above mentioned

change

of variables. For the

particular

^{case N =}

2

^{a =}

1 - 1

^-

1

^our hamiltonians

yield:

^{2 8 2}

Vol. XXXV, ^{n° 1-1981.}

12 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA}

which coincides with the harmonic oscillator hamiltonian

presented by

1 Droz-Vincent

[5 ],

^{if we take :}

KDv

⁼

2 K .

In this case the

position

^and

velocity

^{variables can be}

expressed

^{in a}

particularly simple

^{form :}

where: _y ⁼ _{y2 ,} z ⁼ z2 , r~a ⁼

(- 1)~+ y a

⁼

1,

^2.

Equations (5.6)

coincide with Droz-Vincent’s

expressions.

^{We have}

to

point

^out

that,

when restricted

to ~ _ ~ (TM4)2 ~

^{P -}

( q

1

-

q2)

⁼

0},

not

only x~( q, p)

⁼ but also :

p)

⁼

p~.

On the other

hand,

^{if N} &#x3E; 2 it is no

longer possible

^{to find} suitable values of _(X,

~/~~y

^{such that}

T - - a 1 2 ^{pa 2 .}

^{Because of this it was necessary to gene-}

ralize the

expression

^of

T~

^{in order to describe}

(in

^{a solvable}

way)

^{a model}

involving

^{three or more}

particles.

b~ Singular lagrangian systems.

Most of the

singular lagrangian

systems

existing

ⁱⁿ the literature

present

the constraints :

We are

going

^{to show}

that,

^{when N = 3 and} for suitable values of (x,

{3,

y and

V,

^our hamiltonian model is a

predictive

extension of the

Takabayasi lagrangian

system

[4 ].

^{In other}

words,

^{we are}

going

^to prove

that,

^when restricted to the submanifold

E*,

^the

equations

of motion derived from

our model

yield

^the

equations

of motion of

Takabayasi (*).

Each

integral

^{of our}

generalized

^Hamilton

equations (4.2)

^{is a N-sub-}

manifold

parametrized by

Ti, ...,TN which intersects E* in a curve.

This is the world line of the system ^{on L*. It} is immediate to show that the derivative

a/~~,

^is tangent ^to

E*,

^while

(A

⁼

2,

^...,

N)

^{are not.}

Therefore, ~?

^{is a}

good

parameter ^{to describe the} evolution of _{the sys-}

tem on E*. Hereafter we shall write instead of :E* and we shall consider a

potential

^{of the form}

V(zA)-we

^{have to realize that zA =}

zA

on E*.

(*) The same can be done easily ^{for the Kalb and Van Alstine} Lagrangian [2]

which has the same structure as (5. 10] ^{with N = 2.}

Annales de l’Institut Henri Poincaré-Sectíon A

The

equations

^{of motion}

(4.2)

restricted to E*

yield :

from which we obtain the

following

second order differential

equations :

Let us now consider the

Takabayasi Lagrangian [4 ] :

where

(’)

^{means the} derivative referred to any Poincare invariant _para- meter T and ~’11 is the

projection

^of

perpendicular

^to the relative coor-

dinates _z2 and _z3.

In

fact,

the model

presented

^{in ref.}

[4] ]

^{does not exhibit the cross}

term

(z2, z3).

^We have added it in order to maintain the invariance under

particle interchange.

^{It is immediate to} prove that there is a linear trans- formation which

changes

^the

Takabayasi lagrangian

into the form

(5.10).

From the

lagrangian (5.10)

^we derive the

equations

of motion :

which hold on the constrained submanifold

= 0, PyA = 0, A = 2,3}.

Here pl1

and yA

^{are the}

conjugated

^{momenta of Xl1 and}

zA:

From eq.

[3] ] [8] ] [P] ] [77] ] [72] we

^see

easily

^that both models coincide if we take :

Vol. XXXV, ^{n° 1-1981.}

14 V. IRANZO, ^J. LLOSA, ^F. MARQUES ^{AND A. MOLINA} and

where and C are numerical constants and it is clear that is ^an

integral

of motion. We have to

point

^{out that} eq.

(5.14)

^{is the}

expression

of

U(zA) given by Takabayasi

^{in order to obtain a free}

particle

system when V ⁼ 0.

c)

^Free

particle system.

When V ⁼ 0 the

equations

^{of motion}

(4 . 2)

^{read :}

Hence,

^the

coordinates q a depend linearly

^on 03C4b and the momenta

p 03B1

^are

integrals

^{of motion.}

We have to

point

^{out that the} relation between

p~ and qa

^{is not the usual}

simple

^{one :}

qQ = p~ .

This is due to the fact that we have had to

give

up the usual free

particle

form

of T - - 1 2 a

⁼

^1,

^{..., N.}

Nevertheless,

^{in terms} of the

position

^and

velocity

^{variables we}

can see

immediately

^from eq.

(2.10) (4.10) (4.11)

^and

(5.15)

^{that :}

where oa and oa

^{are the}

^positions

and velocities at T 1 = ^{... =} TN ⁼ 0.

6. CONCLUSION AND OUTLOOK

In earlier models with interaction-at-a-distance

[2] ] [3] ] [4] ] [5] two

^or

three

particle

systems ^have been treated

separately. They

^{neither can be}

generalized

^to

N-particle

systems ^nor

permit

^{to deal with}

quark

interaction for mesons and

baryons

^{in a unified} way. Here ^we have

presented

^a

family

of

N-particle

systems which includes the earlier

quoted

^{ones when N = 2}

or N ⁼ 3 it is a

predictive

extension of some of them

[2] ] [3] ] [4] and

coincides with the

predictive

Droz-Vincent’s model

[5 ].

Many

^{of those models are}

incomplete

^{because the} relation between the

position

^and

velocity

variables and the canonical ones is not

specified.

Hence, ^the

physical meaning

^{of the} coordinates involved remains unknown.

l’Institut l’Institut Poincaré-Section A

We have

carefully

discussed this

question

and the above mentioned rela- tion has been

exactly

derived for our

family

^of systems.

Owing

^{to the}

generality

^{of the}

potential V,

^{our model}

permits

^{to describe}

different kinds of interaction. We have

applied

^{it to the}

particular

^{case of}

a harmonic oscillator

potential

^{since it is}

specially interesting

^{in order}

to describe the

quark

interaction.

In a future work we shall

quantize

the model and we shall _compare the mass-spectrum ^{and other}

properties

^{with the}

phenomenological

^data

obtained from hadrons.

[1] ^L. BEL, A. SALAS and J. M. SÁNCHEZ, Physical Review, ^t. D7, 1973, p. 1099.

[2] M. KALB and P. VAN ALSTINE, Yale Report COO-3075-146 (june 1976).

[3] D.

DOMINICI, ^J. GOMIS and G. LONGHI, Il Nuovo Cimento, ^t. 48B, 1978, p. 152.

[4] ^T. TAKABAYASI, Prog. of Theor. Phys., ^t. 58, 1977, p. 1299.

[5] ^Ph. DROZ-VINCENT, Ann. Inst. H. Poincaré, t. 28A, 1977, p. 407.

[6] ^{H. LEUTWYLER and J.} STERN, ^Ann. of Physics, ^t. 112, 1978, p. 94.

[7] ^{J. A.} LLOSA, F. MARQUÉS and A. MOLINA, Ann. Inst. H. Poincaré, t. 32A, 1980,

p. 303.

[8] ^F. MARQUÉS, Doctoral Thesis. Universitat de Barcelona (september 1980).

[9] H. LEUTWYLER in ^« Quantum Cromodynamics », Lecture Notes in Physics,

t. 118, Springer-Verlag, 1980, p. 121.

[10] ^J. GASSER, ^H. LEUTWYLER, J. JERSAK and J. STERN, ^Berna Preprint, ^1979.

[11] ^{P. A. M.} DIRAC, ^Rev. of Modern Physics, ^t. 21, 1949, p. 392.

[12] ^{D. G.} CURIE, ^T. F. JORDAN and E. C. G., Sudarshan, ^Review of ^Modern Physics,

t. 35, 1963, p. 350.

[13] ^H. LEUTWYLER, ^{Il Nuovo} Cimento, t. 37, 1965, p. 556.

[14] ^Ph. DROZ-VINCENT, Physica Scripta, ^{t. 2, 1970,} p. 129.

[15] ^L. BEL, Ann. Inst. H. Poincaré, t. 14, 1971, p. 189.

[16] ^{L. BEL and J. MARTIN,} Ann. Inst. H. Poincaré, ^t. 22, 1975, _{p. 173.}

[17] L. BEL in Differential Geometry ^and Relativity. ^{Éd. Cohen} and Flato. Dor-

drecht, 1976, p. 197.

[18] ^C. BONA, Doctoral Thesis. Universitat Autonoma Barcelona (june 1980).

[19] ^R. LAPIEDRA, A. MOLINA, ^J. of Math. Phys., ^t. 20, 1979, p. 1309.

[20] ^R. LAPIEDRA, ^F. MARQUÉS and A. MOLINA, ^{J. Math.} Phys., ^t. 20, 1979, p. 1316.

[21] ^{R. P.} FEYNMAN, M. KISLINGER and F. RAVNDAL, Physical Review, ^t. D3, 1971, p. 2706.

[22] ^{Y. S. KIM and M. E. Noz,} Physical Review, t. D8, 1973, p. 3521.

(Manuscril reçu le 17 fevrier 1981)